Chapter 8: Angular momentum

8.2 Conservation of angular momentum

Just like linear momentum, if there is no external influence* acting on an object then angular momentum is conserved:

\[
L_i = L_f \tag{8.2}
\]

A very important result of this describes how a body’s angular speed changes as mass comes closer to the axis of rotation. Recall from section 6.3 that as mass moves towards the axis of rotation, the moment of inertia decreases. Unless something external to the object is acting on it, the angular velocity must increase in order for angular momentum \(\left(L = I\omega\right)\) to remain constant. This is how figure skaters, competitive divers, and gymnasts are able to control how quickly they spin around.

Example

You hold a 2 kg dumbbell in each hand and sit on a stool that spins without friction. When the masses are 80 cm from the center of your body (the axis of rotation as you spin), your angular velocity is 4 rad/s. When you pull your arms in so that they are only 25 cm from the center of your body, what is your new angular speed? Neglect the moment of inertia of your stool.

\[
\begin{align*}
L_i &= L_f \\
I_i\omega_i &= I_f\omega_f
\end{align*}
\]

You have two hands, but each hand is the same distance from you and is holding the same mass, so we can write this as

\[
2I_i\omega_i = 2I_f\omega_f
\]

Treat each dumbbell as a point mass, so its moment of inertia is \(I = mr^2\) where \(r\) is the distance from the dumbbell to the axis of rotation.

\[
\begin{align*}
2mr_i^2\omega_i &= 2mr_f^2\omega_f \\
\hookrightarrow \omega_f &= \left(\frac{r_i}{r_f}\right)^2\omega_i \\
&= \left(\frac{80\ \textrm{cm}}{25\ \textrm{cm}}\right)^2(4\ \textrm{rad/s}) \\
&= 40.96\ \textrm{rad/s}
\end{align*}
\]

If angular momentum were not conserved, the universe would be very different—for example, stars would not be able to form, and cats wouldn’t be able to land on their feet.

8.2.1 Star formation

Stars form out of huge clouds of interstellar gas, which on average have some very slow angular speed. Some spots of the cloud are more densely packed than others. Over long periods of time, hundreds of thousands of years, this extra mass pulls on the other parts of the cloud. Any two objects with mass exert gravitational forces on each other; here on Earth we don’t notice because of the overwhelmingly large mass of the earth itself (we’ll talk more about this interaction when we discuss gravitation in chapter 9).

Particles are pulled to the center—which means r gets smaller. In order for angular momentum to be conserved, v must increase. You end up with lots of particles, moving very quickly, in close proximity. They rub against each other and generate lots of heat. Eventually (over millions of years), there is enough heat generated for hydrogen atoms to fuse together, which releases heat, and you have now created a star. The sun—and you—would not be here if angular momentum was not conserved!

Of course this brief explanation glosses over some key pieces, such as why they cloud is generally rotating in the first place, or why some spots are more densely packed than others, or why fusion releases heat. If you are interested in these details, I highly recommend taking an Astronomy class.

8.2.2 Cats

It is a well-known fact that cat’s land on their feet—this is known as the cat righting reflex. A cat can be suspended upside down, and somehow rotate around so that it’s feet are below it, without anything external to the cat influencing it’s motion. In other words, the cat can go from not rotating, to rotating, to not rotating again—a clear violation of conservation of angular momentum!

Cats are very flexible. When falling, a cat will rotate its front and back halves independently, and it uses its front and back legs to adjust the moment of inertia of each end. It will pull one set of legs underneath itself, lowering the moment of inertia of that end of the cat and allowing that end to rotate faster. At the same time, it extends the other set of legs for the opposite effect: this end of the cat has a larger moment of inertia and can therefore rotate slower while having the same angular momentum.

So, the cat starts out with zero angular momentum (not rotating). The cat can rotate its front end and back end in opposite directions, and by controlling the moment of inertia of each end it can rotate them at different rates. Since angular momentum can have a positive or negative value, depending on direction of rotation, the cat is able to maintain a total angular momentum of zero. The cat’s angular momentum is conserved!

Here is excellent video on this topic, including some good slow-motion footage:

Humans can train to perform similar stunts. Athletes performing a highdive, for example, control their rotation by tucking into a ball to rotate faster, then extending their arms and legs to slow their rotation before they reach the water.

Practice


*See chapter 11 for details